DR/DZ equivalence conjecture and tautological relations

DR/DZ EQUIVALENCE CONJECTURE AND TAUTOLOGICAL RELATIONS

arXiv:1705.03287v1 [math.AG] 9 May 2017

´ EMY ´ ´ E, ´ AND PAOLO ROSSI ALEXANDR BURYAK, JER GUER Abstract. In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/DubrovinZhang equivalence conjecture introduced in [BDGR16a]. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus 0 and 1 and also when first pushed forward from Mg,n+m to Mg,n and then restricted to Mg,n , for any g, n, m ≥ 0. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for g ≤ 2. As an application we find a new formula for the class λg as a linear combination of dual trees intersected with kappa and psi classes, and we check it for g ≤ 3.

Contents 1. Introduction Acknowledgements 2. Tautological relations 2.1. Tautological ring of Mg,n 2.2. Double ramification cycle and the definition of the A-class 2.3. Definition of the B-class and the main conjecture 3. DR/DZ equivalence conjecture and the new tautological relations 3.1. Dubrovin-Zhang hierarchy 3.2. Double ramification hierarchy 3.3. Strong DR/DZ equivalence conjecture 3.4. From intersection numbers to cohomology classes 4. Further structure of the relations 4.1. Formulas with the double ramification cycles 4.2. One-point case 4.3. String equation 4.4. Reduction of the conjecture 4.5. Dilaton equation 4.6. Validity of the conjecture on Mg,n 4.7. New expression for λg 5. Restricted set of relations Appendix A. Proof of the restricted genus 2 relations A.1. Relation A2d = Bd2 2 A.2. Relation A22,1 = B2,1 2 A.3. Relation A21,1,1 = B1,1,1 2 2 and A22,2 = B2,2 A.4. Relations A23,1 = B3,1 2 2 A.5. Relation A2,1,1 = B2,1,1 2 A.6. Relation A21,1,1,1 = B1,1,1,1 References

1

2 3 3 4 5 9 11 11 12 13 14 16 16 17 18 23 24 29 30 31 33 33 34 35 38 39 40 43

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A. Buryak, J. Gu´er´e, P. Rossi

1. Introduction A cohomological field theory (CohFT) cg,n is a family of cohomology classes on the moduli spaces Mg,n of genus g stable curves with n marked points (parameterized by n tensor copies of a vector space) which satisfy certain compatibility axioms with respect to the natural morphisms among different moduli spaces. They were introduced by Kontsevich and Manin [KM94] to axiomatize the properties of Gromov-Witten classes for a given smooth projective variety, but have since then also proved to be a powerful probe for the cohomology and Chow rings of Mg,n itself, and their tautological subrings in particular [PPZ15, Jan15, JPPZ16]. Recall that the tautological rings R∗ (Mg,n ), for g, n ≥ 0 satisfying 2g − 2 + n > 0, are the smallest Q-subalgebras of H ∗ (Mg,n , Q) closed under pushforward along the morphisms forgetting marked points and gluing two marked points together to form a node. R∗ (Mg,n ) is much smaller than the full cohomology ring, but still has a rich structure and contains most of the natural and geometrically interesting classes. The ring structure of R∗ (Mg,n ), however, is not yet completely under control. We know a system of additive generators, the so-called strata algebra, formed by basic classes which are represented by the closure of the loci of curves with fixed dual stable graph intersected with a given monomial in kappa and psi classes. The product of basic classes is explictly described, but the full system of relations is still unknown, although Pixton has found a large set of relations that is conjectured to be complete, see [PPZ15]. In this paper we present a new family of conjectural relations in the form of an equality between two families of tautological classes. We denote these classes in R∗ (Mg,n ) by Agd1 ,...,dn P and Bdg1 ,...,dn , where the n integer non-negative parameters d1 , . . . , dn satisfy 2g − 1 ≤ di ≤ 3g − 3 + n. Their precise definition is given in Sections 2.2 and 2.3 respectively, but here we stress that they can be described as two different linear combinations of stable trees with psi classes at the half-edges and, moreover, for the A-classes only, a double ramification cycle times the Hodge class λg is attached at each vertex. The motivation for this conjecture comes from the study of the double ramification (DR) hierarchy, an integrable system of Hamiltonian PDEs associated to a CohFT and involving the geometry of the DR cycle, introduced by the first author in [Bur15] and further studied in [BR16a, BR16b, BG16, BDGR16a, BDGR16b] (see also [Bur17, Ros17] for a review). In [BDGR16a], sharpening a conjecture from [Bur15], it was conjectured that (the logarithm of) the tau function of (a particular solution of) the DR hierarchy coincides with the reduced potential of the CohFT. The reduced potential is obtained from the full potential, i.e. the generating series of the intersection numbers of the CohFT with monomials in the psi classes, by an explicit procedure, also described in [BDGR16a], which only depends on the potential itself and which ultimately forgets part of the information. In case the CohFT is semisimple (a technical condition on its genus 0 part), the conjecture translates into a statement about the relation between the DR hierarchy and the Dubrovin– Zhang hierarchy, another, more classical, construction associating an integrable system to a semisimple CohFT for which we have the Witten-type result that (the logarithm of) the tau function of (a special solution of) the DZ hiearchy coincides with the potential of the CohFT. In this case the strong DR/DZ equivalence conjecture states that the two hierarchies are related by a normal Miura transformation, i.e. a change of coordinates preserving the taustructure, and hence acting in particular on the tau-functions. This action on the tau-functions precisely corresponds to the reduction procedure described above for the potential of the CohFT.

DR/DZ equivalence conjecture and tautological relations

3

As we have seen, the DR/DZ equivalence conjecture is about intersection numbers, not cohomology classes. However in Section 3 we show how the coefficients of the two involved generating series, the (logarithm of the) DR tau-function and the reduced potential of the CohFT, are the intersection numbers of the CohFT with two different families of cohomology classes. These two families are precisely the A- and B-classes above. So the DR/DZ equivalence conjecture states that the intersection numbers of the A- and B-classes with any (possibly non tautological) CohFT are equal: Z Z g Ad1 ,...,dn cg,n = Bdg1 ,...,dn cg,n . Mg,n

Mg,n

This motivates us to conjecture that it is the A- and B-classes themselves to be equal: Agd1 ,...,dn = Bdg1 ,...,dn . In the rest of the paper we work towards the proof of such conjecture. In Section 4 we prove the string and dilaton equations for both A- and B-classes, establishing that their behaviour upon pullback and pushforward along the morphism π : Mg,n+1 → Mg,n that forgets the last marked point is the same. The string equation allows us to prove that the conjecture is true if and only if it is true when all the parameters d1 , . . . , dn are strictly positive. This in turn yields a full proof of the conjecture in genus 0 and genus 1. The dilaton equation is used to show that the relations in R∗ (Mg,n ) obtained by pushing forward our conjectural relations from R∗ (Mg,n+m ) to R∗ (Mg,n ) and then restricting them to R∗ (Mg,n ) are valid. This is what we mean by saying that the conjecture is valid on Mg,n . We then show that our relations imply in particular a new expression for the top Chern class of the Hodge bundle λg as a linear combination of basic classes whose dual graph is a tree (with psi and kappa classes). No expression of this type for λg was known before. We check its validity for g ≤ 3. Finally, in Section 5 we show that, for semisimple CohFTs, the DR/DZ equivalence conjecture P actually depends on just a subset of our conjectural relations, namely the ones for which di = 2g and di > 0. This means that the number of relations one needs to check is finite in each genus, and equal to the number of partitions of 2g. In the appendix we check this finite subset of relations for g = 2 thereby proving the strong DR/DZ equivalence conjecture in genus g ≤ 2 for any semisimple CohFT. Acknowledgements. We would like to thank Boris Dubrovin, Rahul Pandharipande, Sergey Shadrin and Dimitri Zvonkine for useful discussions. A. B. was supported by Grant ERC-2012AdG-320368-MCSK in the group of Rahul Pandharipande at ETH Zurich and Grant RFFI-1601-00409. J. G. was supported by the Einstein foundation. P. R. was partially supported by a Chaire CNRS/Enseignement superieur 2012-2017 grant. Part of the work was completed during the visit of J. G. and P. R. of the Forschungsinstitut f¨ ur Mathematik at ETH Z¨ urich in 2017. 2. Tautological relations In this section we present our conjectural tautological relations.

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A. Buryak, J. Gu´er´e, P. Rossi

2.1. Tautological ring of Mg,n . Here we fix notations concerning tautological cohomology classes on Mg,n . We will use the notations from [PPZ15, Sections 0.2 and 0.3]. Recall that for any stable graph Γ we have the associated moduli space Y MΓ := Mg(v),n(v) v∈V (Γ)

and the canonical morphism

ξΓ : MΓ → Mg(Γ),|L(Γ)| . Recall [PPZ15] that given numbers xi [v], y[h] ≥ 0, i ≥ 1, v ∈ V (Γ), h ∈ H(Γ), we can define a basic cohomology class on MΓ by Y Y Y y[h] γ= κi [v]xi [v] · ψh ∈ H ∗ (MΓ , Q), (2.1) v∈V (Γ) i≥1

h∈H(Γ)

where κi [v] is the i-th kappa class on Mg(v),n(v) and ψh is the psi-class on Mg(v(h)),n(v(h)) . A cohomology class on Mg,n of the form ξΓ∗ (γ), where Γ is a stable graph of genus g with n legs and γ is a basic class on MΓ , will be called a basic tautological class. Denote by R∗ (Mg,n ) the subspace of H ∗ (Mg,n , Q) spanned by all basic tautological classes. The subspace R∗ (Mg,n ) is closed under multiplication and is called the tautological ring of the moduli space of curves. Let Ri (Mg,n ) := R∗ (Mg,n ) ∩ H 2i (Mg,n , Q). Denote by Mct g,n ⊂ Mg,n the moduli space of curves of compact type and by Mg,n ⊂ Mg,n the moduli space of smooth curves. We will use the notations Ri (Mct ) := Ri (Mg,n ) ct , Ri (Mg,n ) := Ri (Mg,n ) . g,n

Mg,n

Mg,n

Linear relations between basic tautological classes are called tautological relations. The class ξΓ∗ (1) ∈ R|E(Γ)| (Mg(Γ),|L(Γ)| ) will be called a boundary stratum. We will represent a basic tautological class ξΓ∗ (γ) on Mg(Γ),|L(Γ)| by a picture of the graph Q Γ where we put the monomial i≥1 κi [v]xi [v] next to each vertex v and the power of the psiy[h] class ψh next to each half-edge h. For example, we have the following well-known formulas: 1

ψ1 =

3 0

2

0

∈ R1 (M0,4 ),

4

1 1 0 ∈ R1 (M1,1 ), 24 2i where we denote by λi ∈ H (Mg,n , Q) the i-th Chern class of the Hodge vector bundle over Mg,n . It is well-known that the class λi is tautological (see e.g. [FP00]). Suppose Γ1 and Γ2 are two stable graphs, both of genus g and with n legs. They are called isomorphic, if there exists a pair f = (f1 , f2 ) of set isomorphisms f1 : V (Γ1 ) → V (Γ2 ) and f2 : H(Γ1 ) → H(Γ2 ) that preserve all the additional structure of the stable graphs. Suppose γ1 and γ2 are two basic classes on the spaces MΓ1 and MΓ2 respectively: Y Y Y Y Y Y y [h] y [h] γ1 = κi [v]x1,i [v] · ψh1 , γ2 = κi [v]x2,i [v] · ψh2 . λ1 =

v∈V (Γ1 ) i≥1

h∈H(Γ1 )

v∈V (Γ2 ) i≥1

h∈H(Γ2 )

We will say that the pairs (Γ1 , γ1 ) and (Γ2 , γ2 ) are combinatorially equivalent, if there exists a pair of maps f = (f1 , f2 ), f1 : V (Γ1 ) → V (Γ2 ), f2 : H(Γ1 ) → H(Γ2 ), that defines an isomorphism between the stable graphs Γ1 and Γ2 and also satisfies the properties x1,i [v] = x2,i [f1 (v)],

for any i ≥ 1 and v ∈ V (Γ1 ),

y1 [h] = y2 [f2 (h)],

for any h ∈ H(Γ1 ).

Obviously, if the pairs (Γ1 , γ1) and (Γ2 , γ2 ) are combinatorially equivalent, then ξΓ1 ∗ (γ1 ) = ξΓ2 ∗ (γ2 ).

DR/DZ equivalence conjecture and tautological relations

5

Consider the set of stable graphs of genus g with n legs. Suppose I is a subset of {1, 2, . . . , n}. The symmetric group S|I| acts on our set of stable graphs by permutations of markings from the set I. This gives an S|I| -action on the set of pairs (Γ, γ), where Γ is a stable graph and γ is a basic class on MΓ . Let us fix some stable graph Γ and a basic class γ. The sum of the basic tautological classes corresponding to combinatorially non-equivalent pairs in the S|I| -orbit of the pair (Γ, γ) will be represented by the picture corresponding to the class ξΓ∗ (γ), where we erase the labels from the set I. Let us give two examples in order to illustrate this rule: 0

1

0

1

1

ψ

= =

1

0

1

1

2

1

1

3

+

ψ

3 1

+

0

2

4 1

0 1

1

2 1

ψ

1 3

+

0 2

4

0 1

1 4

,

0 2

3

.

3

As another useful example, we can write the topological resursion relations in genus 0 and 1: (2.2)

ψ1 =

X

2 1

0

3 ... ... |{z} |{z}

i+j=n−3 i≥1, j≥0

(2.3)

ψ1 =

1 24

1

0

∈ R1 (M0,n ),

n ≥ 4,

i legs j legs

+

0 ... |{z}

n − 1 legs

X

i+j=n−1 i≥1, j≥0

1

0

1

∈ R1 (M1,n ).

... ... |{z} |{z} i legs j legs

By stable tree we mean a stable graph Γ with the first Betti number b1 (Γ) equal to zero. Suppose Γ is a stable tree. Let H e (Γ) := H(Γ)\L(Γ). A path in Γ is a sequence of pairwise distinct vertices v1 , v2 , . . . , vk ∈ V (Γ), vi 6= vj for i 6= j, such that for any 1 ≤ i ≤ k − 1 the vertices vi and vi+1 are connected by an edge. For a vertex v ∈ V (Γ) define a number r(v) by r(v) := 2g(v) − 2 + n(v). Denote by STm g,n the set of stable trees of genus g with m vertices and with n legs marked by numbers 1, . . . , n. For a stable tree Γ ∈ STm g,n denote by li (Γ) the leg in Γ that is marked by i. For a leg l ∈ L(Γ) denote by 1 ≤ i(l) ≤ n its marking. A stable rooted tree is a pair (Γ, v1 ), where Γ is a stable tree and v1 ∈ V (Γ). The vertex v1 is called the root. Denote by H+ (Γ) the set of half-edges of Γ that are directed away from the root v1 . Clearly, L(Γ) ⊂ H+ (Γ). Let H+e (Γ) := H+ (Γ)\L(Γ). A vertex w is called a descendant of a vertex v, if v is on the unique path from the root v1 to w. Note that according to our definition the vertex v is a descendant of itself. Denote by Desc[v] the set of all descendants of v. A vertex w is called a direct descendant of v, if w ∈ Desc[v], w 6= v and w and v are connected by an edge. In this case the vertex v is called the mother of w. 2.2. Double ramification cycle and the definition of the A-class. Consider integers a1 , . . . , an such that a1 + . . . + an = 0. The double ramification cycle DRg (a1 , . . . , an ) is a cohomology class in H 2g (Mg,n , Q). If not all of ai ’s are equal to zero, then the restriction DRg (a1 , . . . , an )|Mg,n can be defined as the Poincar´e dual to the locus of pointed smooth P curves (C, p1, . . . , pn ) satisfying OC ( ni=1 ai pi ) ∼ = OC , and we refer the reader, for example, to [BSSZ15] for the definition of the double ramification cycle on the whole moduli space Mg,n . We will often consider the Poincar´e dual to the double ramification cycle DRg (a1 , . . . , an ). It is

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A. Buryak, J. Gu´er´e, P. Rossi

an element of H2(2g−3+n) (Mg,n , Q) and, abusing our notations a little bit, it will also be denoted by DRg (a1 , . . . , an ). The double ramification cycle DRg (a1 , . . . , an ) is a tautological class on Mg,n [FP05]. A was derived in [Hai13, MW13]: simple explicit formula for the restriction DRg (a1 , . . . , an )|Mct g,n g  n

(2.4)

DRg (a1 , . . . , an )|Mct g,n

1 X a2i ψi 1 =  − g!  i=1 2 2

X

I⊂{1,...,n} |I|≥2

a2I δ0I

1 − 4

X

g−1 X

I⊂{1,...,n} h=1

 a2I δhI   ,

where for a subset I ⊂ {1, 2, . . . , n} and a number 0 ≤ h ≤ g we use the following notations: X aI := ai , i∈I

δhI

:=

h

h′

∈ R1 (Mg,n ),

I c := {1, 2, . . . , n}\I,

h′ := g − h.

... ... |{z} |{z} I

Ic

Formula (2.4) is usually referred as Hain’s formula. It implies that the class DRg (a1 , . . . , an )|Mct g,n is a polynomial in the variables a1 , . . . , an homogeneous of degree 2g. Since λg |Mg,n \Mct = 0, g,n we obtain that the class λg DRg (a1 , . . . , an ) ∈ R2g (Mg,n ) is a polynomial in a1 , . . . , an homogeneous of degree 2g. The full double ramification cycle is also polynomial, but not necessarily homogeneous [JPPZ16]. The following properties of the double ramification cycle will be useful for us. Let πi : Mg,n+1 → Mg,n be the forgetful map that forgets the i-th marked point. Then ∗ DRg (a1 , . . . , an , 0) = πn+1 DRg (a1 , . . . , an ).

Let π : Mg,n+g → Mg,n be the forgetful map that forgets the last g marked points. Then we have [BSSZ15, Example 3.7] (2.5)

π∗ DRg (a1 , . . . , an+g ) = g!a2n+1 · · · a2n+g [Mg,n ].

It is also useful to remember that (see e.g. [JPPZ16]) DRg (0, 0, . . . , 0) = (−1)g λg ∈ Rg (Mg,n ). We will denote by DRg (a1 , . . . , aei , . . . , an ) the class πi∗ DRg (a1 , . . . , an ) ∈ Rg−1 (Mg,n−1 ). Recall the following important divisibility property. Lemma 2.1 ([BDGR16a]). Let g, n ≥ 1. Then the polynomial class  X  DRg − ai , a1 , a2 , . . . , aen ∈ Rg−1 (Mct g,n ) Mct g,n

is divisible by a2n .

Consider a stable tree Γ ∈ STm g,n and integers a1 , . . . , an such that a1 + . . . + an = 0. To each half-edge h ∈ H(Γ) we assign an integer a(h) in such a way that the following conditions hold: a) If h ∈ L(Γ), then a(h) = ai(h) ; b) If h ∈ H e (Γ), then a(h) + a(ι(h)) = P0; c) For any vertex v ∈ V (Γ), we have h∈H[v] a(h) = 0.

Clearly, such a function a : H(Γ) → Z exists and is uniquely determined by the numbers a1 , . . . , an . For each moduli space Mg(v),n(v) , v ∈ V (Γ), the numbers a(h), h ∈ H[v], define the double ramification cycle
DRg(v) AH[v] ∈ Rg(v) (Mg(v),n(v) ).

DR/DZ equivalence conjecture and tautological relations

7

Here AH[v] denotes the list a(h1 ), . . . , a(hn(v) ), where {h1 , . . . , hn(v) } = H[v]. If we multiply all these cycles, we get the class Y
DRg(v) AH[v] ∈ H 2g (MΓ , Q). v∈V (Γ)

We define a class DRΓ (a1 , . . . , an ) ∈ Rg+m−1 (Mg,n ) by   Y
DRΓ (a1 , . . . , an ) := ξΓ∗  DRg(v) AH[v]  . v∈V (Γ)

Clearly, the class

λg DRΓ (a1 , . . . , an ) ∈ R2g+m−1 (Mg,n ) is a polynomial in a1 , . . . , an homogeneous of degree 2g. P Suppose now that a1 , . . . , an are arbitrary integers and let a0 := − ni=1 ai . Consider the set of stable trees STm g,n+1 . It would be convenient for us to assume that the legs of stable trees m from STg,n+1 are marked by 0, 1, . . . , n. Let Γ ∈ STm g,n+1 be an arbitrary stable tree. Consider it as a rooted tree with the root v1 (Γ) := v(l0 (Γ)). As above, the numbers a0 , a1 , . . . , an define a function a : H(Γ) → Z. Define a coefficient a(Γ) by    Y Y r(v) . P a(Γ) :=  a(h)  v) v e∈Desc[v] r(e e h∈H+ (Γ)

v∈V (Γ)

Let π : Mg,n+1 → Mg,n be the forgetful map that forgets the first marked point. Define a class eg,m (a1 , . . . , an ) ∈ R2g+m−2 (Mg,n ) by A X eg,m (a1 , . . . , an ) := a(Γ)λg π∗ DRΓ (a0 , a1 , . . . , an ). A Γ∈STm g,n+1

We know that this class is a polynomial in a1 , . . . , an homogeneous of degree 2g + m − 1. Note eg,1 (a1 , . . . , an ) is actually very simple: that the expression for the class A eg,1 (a1 , . . . , an ) = λg DRg (ae0 , a1 , . . . , an ). A eg,m (a1 , . . . , an ) is divisible by Pn ai . Lemma 2.2. The polynomial class A i=1

Proof. If m = 1, then the lemma follows from Lemma 2.1. Suppose m ≥ 2 and a0 = − 0. We have to prove that eg,m (a1 , . . . , an ) = 0. A Consider a stable tree Γ ∈ STm g,n+1 . If g(v1 (Γ)) ≥ 1, then, again by Lemma 2.1,

Pn

i=1

ai =

λg π∗ DRΓ (0, a1 , . . . , an ) = 0.

If g(v1 (Γ)) = 0, then π∗ DRΓ (0, a1 , . . . , an ), unless n(v1 (Γ)) = 3. We obtain X eg,m (a1 , . . . , an ) = (2.6) a(Γ)λg π∗ DRΓ (0, a1 , . . . , an ) . A Γ∈STm g,n+1 g(v1 (Γ))=0 n(v1 (Γ))=3

Let us define certain maps m STm−1 g,n → {Γ ∈ STg,n+1 |g(v1 (Γ)) = 0, n(v1 (Γ)) = 3}.

Note that we mark the legs of stable trees from STm−1 g,n by 1, . . . , n and the legs of stable trees m m−1 from STg,n+1 by 0, 1, . . . , n. Let Γ ∈ STg,n . Choose a leg l ∈ L(Γ). Suppose that it is marked by number 1 ≤ i ≤ n. Let us attach to the leg l a new vertex of genus 0 with two legs marked by numbers 0 and i. Denote the resulting stable tree by Φl (Γ) ∈ STm g,n+1 . Similarly, if we choose an edge e ∈ E(Γ), then we can break this edge and insert a genus 0 vertex with one leg

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A. Buryak, J. Gu´er´e, P. Rossi

marked by 0. Denote the resulting stable tree by Φe (Γ) ∈ STm g,n+1 . Using these operations, we can rewrite formula (2.6) in the following way:   X X X eg,m (a1 , . . . , an ) =  A a(Φl (Γ)) + a(Φe (Γ)) λg DRΓ (a1 , . . . , an ). Γ∈STm−1 g,n

l∈L(Γ)

e∈E(Γ)

We see that it is sufficient to prove that for any stable tree Γ ∈ STm−1 g,n we have the identity X X (2.7) a(Φl (Γ)) + a(Φe (Γ)) = 0. l∈L(Γ)

e∈E(Γ)

We prove (2.7) by induction on m. It will be convenient for us to assume that the genus g(v) of a vertexP v ∈ V (Γ) can be a rational number such that 2g(v) − 2 + n(v) > 0. So the total genus g = v∈V (Γ) g(v) can also be rational. If m = 2, then X

a(Φl (Γ)) =

n X i=1

l∈L(Γ)

−ai = 0. 2g − 1 + n

Suppose m ≥ 3. Choose a vertex v ∈ V (Γ) such that |H[v]\L[v]| = 1. Let h be the unique half-edge from the set H[v]\L[v]. Denote h′ := ι(h),

v ′ := v(h′ ),

r ′ := r(v ′),

r := r(v),

R := 2g − 2 + n.

Denote by e the edge of Γ corresponding the pair of half-edges (h, h′ ). Let us erase the vertex v together with all half-edges incident to it. Then the half-edge h′ becomes a leg. Let us denote it by l′ and mark by n + 1. Finally, let us increase the genus of the vertex v ′ by 2r . As a result, we ′ ′ get a stable tree from STm−2 g+ r2 ,n−|L[v]|+1 that we denote by Γ . Note that the legs of Γ are marked by the numbers i(l), l ∈ L(Γ)\L[v], and n + 1. We want to apply the induction assumption to the tree Γ′ . Naturally, we assign to a leg l ∈ L(Γ′ ) the number ai(l) , if l 6= l′ , and the number P a(h′ ) = el∈L[v] a(e l), if l = l′ . It is not hard to see that X

a(Φl (Γ)) = (−a(h′ ))

l∈L[v]

a(Φe (Γ)) = a(h′ )

rr ′ a(Φl′ (Γ′ )), (R − r)(r + r ′ )

r′R a(Φl′ (Γ′ )). (R − r)(r + r ′ )

It is also easy to see that for any leg l ∈ L(Γ′ ), l 6= l′ , and for any edge e′ ∈ E(Γ′ ) we have a(Φl (Γ)) =

r′ a(h′ )a(Φl (Γ′ )), ′ r+r

Therefore, we obtain X

l∈L(Γ)

a(Φl (Γ)) +

X

′

a(Φe′ (Γ)) =

e′ ∈E(Γ)

a(Φe′ (Γ)) = 

r a(h′ )  r + r′

X

l∈L(Γ′ )

r′ a(h′ )a(Φe′ (Γ′ )). ′ r+r

a(Φl (Γ′ )) +

X

e′ ∈E(Γ′ )

a(Φe′ (Γ′ )) = 0,

where the last equality follows from the induction assumption. The lemma is proved. The lemma allows to define a class Ag,m (a1 , . . . , an ) by 1 eg,m (a1 , . . . , an ) ∈ R2g+m−2 (Mg,n ). Ag,m (a1 , . . . , an ) := P A ai

It is a polynomial in a1 , . . . , an homogeneous of degree 2g + m − 2. P Definition 2.3. For any d1 , . . . , dn ≥ 0 such that δ := ni=1 di ≥ 2g − 1 we define Agd1 ,...,dn := Coef ad1 ···adnn Ag,δ−2g+2 (a1 , . . . , an ) ∈ Rδ (Mg,n ). 1





DR/DZ equivalence conjecture and tautological relations

If

P

9

di = 2g − 1, then the formula for Agd1 ,...,dn becomes particularly simple:   X  1 ^ g Ad1 ,...,dn = Coef ad1 ···adnn P λg DRg − ai , a1 , . . . , an . 1 ai

2.3. Definition of the B-class and the main conjecture. Let T be a stable rooted tree with at least n legs, where we split the set of legs in two subsets: - the legs σ1 , . . . , σn corresponding to the markings, - some extra legs, whose set is denoted by F (T ), corresponding to additional marked points that we will eventually forget. We will never call marking an element of F (T ) and let H+em(T ) := H+ (T )\F (T ). There is a natural level function l : V (T ) → N∗ such that the root is of level 1 and if a vertex v is the mother of a vertex v ′ , then l(v ′ ) = l(v) + 1. The total number of levels in T will be denoted by deg(T ) and called the degree of T . It is also convenient to extend the level function to H+em (T ) by taking l(h) := k if the half-edge h is attached to a vertex of level k. We say that T is complete if the following conditions are satisfied: - every vertex has at least one of its descendants with level deg(T ), - all the markings are attached to the vertices of level deg(T ), - each vertex of level deg(T ) is attached to at least one marking, - there are no extra legs attached to the root, - for every vertex except the root there is at least one extra leg attached to it. For a complete tree T define a power function q : H+e (T ) → N by requiring that for a half-edge h ∈ H+e (T ) there is exactly q(h) + 1 extra legs attached to the vertex v which is the direct descendant of h. We say that T is stable if - for every 1 ≤ k ≤ deg(T ), there is at least one vertex v ∈ V (T ) of level k such that v remains stable once we forget all the extra legs, - every vertex of genus 0 with exactly one half-edge h ∈ H+em (T ) attached to it has exactly q(h) + 1 extra legs attached to it, - every vertex of genus 0 with exactly two half-edges h1 , h2 ∈ H+em (T ) attached to it has exactly q(h1 ) + q(h2 ) extra legs attached to it. We say that a stable complete tree T is admissible if for every 1 ≤ k < deg(T ) we have the condition   (2.8)

X

e (T ) h∈H+ l(h)=k

  X  − 2. g(v) q(h) ≤ 2    v∈V (T ) l(v)≤k

We denote by ΩB,g d1 ,...,dn the set of pairs (T, q), where T is an admissible stable complete tree with total genus g and n markings, and q : H+em (T ) → N is the extension of the power function from above defined by q(σi ) := di . We denote by   Y q(h) [T, q] := ξT ∗  ψh  ∈ R∗ (Mg,n+#F (T )) em (T ) h∈H+

and by

e : Mg,n+#F (T ) → Mg,n the map forgetting all the extra legs.

10

A. Buryak, J. Gu´er´e, P. Rossi

Definition 2.4. For any d1 , . . . , dn ≥ 0 with δ := d1 + · · · + dn , we define X (−1)deg(T )−1 e∗ [T, q] ∈ Rδ (Mg,n ). (2.9) Bdg1 ,...,dn = (T,q)∈ΩB,g d ,...,dn 1

Conjecture 2.5. Suppose g ≥ 0, n ≥ 1 and 2g − 2 + n > 0. Then for any d1 , . . . , dn ≥ 0, such P that di ≥ 2g − 1, we have Agd1 ,...,dn = Bdg1 ,...,dn .

(2.10)

Remark 2.6. Let us show how to express the B-class in terms of basic tautological classes. Let T be a stable complete tree with n markings. For a vertex v ∈ V (T ) denote by F [v] the set of extra legs incident to v and by H+em [v] the set of half-edges h ∈ H+em (T ) incident to v. The vertex v will be called strongly stable if it remains stable once we forget all the extra legs. Otherwise, we call it weakly stable. Clearly, the vertex v is weakly stable if and only if g(v) = 0 and |H+em [v]| = 1. The set of all strongly stable vertices of T will be denoted by V ss (T ). For a stable complete tree T denote by st(T ) the stable rooted tree obtained by forgetting all extra legs of T and then contracting all unstable vertices. Clearly, we can identify V (st(T )) = V ss (T ) and we also identify the set H(st(T )) with the set of half-edges h ∈ H(T ) such that v(h) is strongly stable. Suppose π : Mg,n+m → Mg,n is the forgetful map that forgets the last m marked points. Then for any numbers c1 , . . . , cn ≥ 0 we have n

X

π∗ (ψ1c1 · · · ψncn ) =

b1 ,...,bn ≥0 P bi ≤ci P bi +m= ci

Y m! Q ψ bi . (ci − bi )! i=1 i

Using this formula, it is easy to see that equation (2.9) can be rewritten in the following way: Y X Y X |F [v]|! p(h) Q ψh . Bdg1 ,...,dn = (−1)deg(T )−1 ξst(T )∗ (q(h) − p(h))! B,g (T,q)∈Ωd

v∈V (st(T ))

1 ,...,dn

P

h∈H+ [v]

p : H+ [v]→Z≥0 p(h)≤q(h)P p(h)+|F [v]|= q(h)

Let us immediately present some examples of relations (2.10). Consider genus 0. Then it is easy to see that for any d1 , . . . , dn ≥ 0 we have Bd01 ,...,dn = ψ1d1 · · · ψndn . On the other hand, let us compute, for example, A01,0,0,0 . We have  X

 e0,3 (a1 , a2 , a3 , a4 ) =π∗  A 

{i,j,k,l}={1,2,3,4} i 0. Then m X DR di < 2g − 1. hτd1 (eα1 ) · · · τdm (eαm )ig = 0, if i=1

[−2] Proposition 3.2 ([BDGR16a]). There exists a unique differential polynomial P ∈ Abw such that for the power series F red ∈ C[[t∗∗ , ε]], defined by

(3.2)

F red := F + P(w∗∗ , ε)|w∗∗ =(wtop )∗∗ (x,t∗∗ ,ε)|x=0 ,

the correlators hτd1 (eα1 ) · · · τdn (eαn )ired g

:= Coef ε2g

∂ n F red ∂tαd11 · · · ∂tαdnn t∗ =0 ∗

satisfy the following vanishing property: (3.3)

hτd1 (eα1 ) · · · τdn (eαn )ired g

= 0,

if

n X

di < 2g − 1.

i=1

In light of these two results the following conjecture was formulated in [BDGR16a]. Conjecture 3.3 (Strong DR/DZ equivalence). Consider a semisimple cohomological field theory and the associated DZ and DR hierarchies. Then the normal Miura transformation (3.1) defined by the differential polynomial P of Proposition 3.2 maps the DZ hierarchy to the DR hierarchy respecting their tau-structures. As proved in [BDGR16a] this conjecture is equivalent to saying that F red = F DR . This last form of the conjecture can be generalized to arbitrary CohFTs, forgetting about the DZ hierarchy and concentrating on the reduced and DR potentials. Conjecture 3.4 (Generalized strong DR/DZ equivalence). For an arbitrary cohomological field theory we have F DR = F red .

14

A. Buryak, J. Gu´er´e, P. Rossi

3.4. From intersection numbers to cohomology classes. The following result makes the relation between Conjecture 2.5 and Conjecture 3.4 explicit, showing in particular how the first implies the second. even Proposition 3.5. Consider an arbitrary cohomological field theory cg,n : V ⊗n → PH (Mg,n , C). Then for any g, n ≥ 0, 2g − 2 + n > 0, and numbers d1 , . . . , dn ≥ 0 such that di ≥ 2g − 1 we have Z DR (3.4) Agd1 ,...,dn cg,n (eα1 ⊗ · · · ⊗ eαn ), hτd1 (eα1 ) · · · τdn (eαn )ig = M Z g,n hτd1 (eα1 ) · · · τdn (eαn )ired Bdg1 ,...,dn cg,n (eα1 ⊗ · · · ⊗ eαn ). (3.5) g = Mg,n

Proof. In [BDGR16b] the authors proved that for any d ≥ 2g − 1 we have X d1 dn hτd1 (eα1 ) · · · τdn (eαn )iDR g a1 · · · an = d1P ,...,dn ≥0 di =d

Z  X  X 1 =P a(Γ) DRΓ − ai , a1 , . . . , an λg cg,n+1 (e1 ⊗ ⊗ni=1 eαi ) = ai Mg,n+1 Γ∈STd−2g+2 g,n+1 Z Ag,d−2g+2 (a1 , . . . , an )cg,n (⊗ni=1 eαi ) . = Mg,n

Equation (3.4) is proved. Let us prove equation (3.5). The reduced potential F red can be constructed in the following way. Let us recursively construct a sequence of power series F (0,−2) = F, F (1,0) , F (2,0) , F (2,1) , F (2,2) , . . . , F (j,0) , F (j,1) , . . . , F (j,2j−2), . . . ∈ C[[t∗∗ , ε]]. Suppose that a series F (j,k) is already defined. Introduce correlators hτd1 (eα1 ) · · · τdn (eαn )i(j,k) g by ∂ n F (j,k) (j,k) hτd1 (eα1 ) · · · τdn (eαn )ig := Coef ε2g α1 . ∂td1 · · · ∂tαdnn t∗ =0 ∗

(j,k+1)

If k < 2j − 2, then we define the series F by D E(j,k) Y 2j Y X X ε (j,k+1) (j,k) (3.6) ((w top )αdii − δ αi ,1 δdi ,1 )|x=0 . τdi (eαi ) F := F − n! j n≥0 d 1 ,...,dn ≥0 P di =k+1

If k = 2j − 2, then we define the series F (j+1,0) by an analogous formula * n +(j,2j−2) X ε2j+2 Y Y F (j+1,0) := F (j,2j−2) − τ0 (eαi ) (w top )αi |x=0. n! n≥0 i=1 j+1

Recall that (w

top α

) =η

αµ

∂ 2 F . ∂tµ0 ∂t10 t1 7→t1 +x 0

0

The string equation for the potential F , X 1 ∂F ∂F = tαn+1 α + ηαβ tα0 tβ0 + ε2 hτ0 (e1 )i1 , 1 ∂t0 ∂tn 2 n≥0 implies that the function (w top )αn |x=0 has the form

(w top )αn |x=0 = δ α,1 δn,1 + tαn + rnα + O(ε2),

DR/DZ equivalence conjecture and tautological relations

15

P where the power series rnα ∈ C[[t∗∗ ]] doesn’t contain monomials tβb11 · · · tβbmm with bi ≤ n. Clearly, if g ≤ j, then we have the property ( X 2g − 2, if g < j, = 0, if di ≤ hτd1 (eα1 ) · · · τdn (eαn )i(j,k) g k, if g = j.

Define a series F ′ by F ′ := limj→∞ F (j,2j−2) . The series F ′ has the form F ′ = F + P ′ (w top , wxtop , . . . , ε) x=0 P [i] for some non-homogeneous differential polynomial P ′ = i≤−2 Pi′ , Pi′ ∈ Abw . Moreover, we have the property X ∂nF ′ = 0, if di ≤ 2g − 2. Coef ε2g α1 αn ∂t · · · ∂t d1

dn t∗∗ =0

One can notice that the recursive construction, described above, is slightly different from the recursive construction of the reduced potential F red , presented in the proof of Proposition 7.2 in [BDGR16a]. However, using the uniqueness argument given there we can see that F ′ = F red . For a stable complete tree T and 1 ≤ m ≤ deg(T ) let X g(v). gm (T ) := v∈V (T ) l(v)≤m

Before we proceed, let us prove the following simple lemma. Lemma 3.6. Let d1 , . . . , dn ≥ 0, (T, q)P∈ ΩB,g 1 ≤ m < deg(T ). Suppose that d1 ,...,dn and P e (T ) q(h). e gm+1 (T ) = gm (T ) and e∗ [T, q] 6= 0. Then h∈H+ h∈H+ (T ) q(h) > l(h)=m

l(h)=m+1

H+e (T )

Proof. Consider a half-edge h ∈ with l(h) = m and let v := v(ι(h)). We have g(v) = 0 and the P map e forgets all q(h) + 1 extra legs incident to v. Therefore, if v is strongly stable, then h′ ∈H e [v] q(h′ ) > q(h). If v is weakly stable, then |H+e [v]| = 1 and q(h′ ) = q(h), where + h′ ∈ H+e [v]. Since at least one vertex of level m + 1 is strongly stable, the lemma is true.  A stable complete tree T will be called (j, k)-admissible, if for any 1 ≤ m < deg(T ) we have gm (T ) ≤ j and ( X 2gm (T ) − 2, if gm (T ) < j, q(h) ≤ k, if gm (T ) = j. h∈H e (T ) +

l(h)=m

B,g,(j,k)

g,(j,k)

Let Ωd1 ,...,dn := {(T, q) ∈ ΩB,g d1 ,...,dn |T is (j, k)-admissible}. Define a class Bd1 ,...,dn by X P g,(j,k) Bd1 ,...,dn := (−1)deg(T )−1 e∗ [T, q] ∈ R di (Mg,n ). B,g,(j,k) 1 ,...,dn

(T,q)∈Ωd

g,(j,k)

Clearly, Bd1 ,...,dn = Bdg1 ,...,dn , if j > g. In order to prove equation (3.5), it is sufficient to prove that for any pair (j, k) from the sequence (3.7)

(0, −2), (1, 0), (2, 0), (2, 1), (2, 2), . . . , (j, 0), (j, 1), . . . , (j, 2j − 2), . . .

we have (3.8)

hτd1 (eα1 ) · · · τdn (eαn )i(j,k) g

if g > j, or g ≤ j and X

di >

=

Z

Mg,n

(

g,(j,k)

Bd1 ,...,dn cg,n (eα1 ⊗ · · · ⊗ eαn ),

2g − 2, if g < j, k, if g = j.

16

A. Buryak, J. Gu´er´e, P. Rossi

We proceed by induction. Obviously, equation (3.8) is true for (j, k) = (0, −2). Suppose that equation (3.8) is true for a pair (j, k) from the sequence (3.7). Let us check it for the next pair. B,g,(j,k) B,g,(j,k+1) Suppose k < 2j − 2. For any d1 , . . . , dn ≥ 0 we have Ωd1 ,...,dn ⊂ Ωd1 ,...,dn . Using the induction assumption and formula (3.6), we see that it remains to check that (3.9)

X ε2g n! g,n≥0

X

X

(−1)

deg(T )

d1 ,...,dn ≥0 (T,q)∈ΩB,g,(j,k+1) \ΩB,g,(j,k)

=

X

d1 ,...,dn

X

d1 ,...,dn

Z

Mg,n

e∗ [T, q]cg,n (⊗ni=1 eαi )

!

Y

tαdii =

E(j,k) Y Y  ε2j DY τdi (eαi ) ((w top )αdii − δ αi ,1 δdi ,1 )|x=0 − tαdii . n! j ≥0

n≥0 d 1 ,...,dn P di =k+1

B,g,(j,k+1)

B,g,(j,k)

Consider a pair (T, q) ∈ Ωd1 ,...,dn \Ω P d1 ,...,dn such that e∗ [T, q] 6= 0. Then there exists 1 ≤ m < deg(T ) such that gm (T ) = j and h∈H+e (T ) q(h) = k + 1. By Lemma 3.6, m = deg(T ) − 1. l(h)=m

Denote by T ′ the stable rooted tree obtained by erasing all vertices in T of level m + 1 together with half-edges incident to them. Half-edges h ∈ H+e (T ) with l(h) = m become marked legs of T ′ . Clearly, T ′ is a stable complete tree. By Lemma 3.6, the tree T ′ is (j, k)-admissible. Using the induction assumption, we conclude that equation (3.9) is true. This completes the induction step in the case k < 2j − 2. The case k = 2j − 2 is analagous. The proposition is proved. 

4. Further structure of the relations In this section we discuss the structure of the conjectural relations (2.10) in more details. In Section 4.1 we recall the formulas for the intersections of the double ramification cycle with a ψ-class and with a boundary divisor on Mg,n . In Section 4.2 we show that for a fixed g ≥ 1 g all relations Agd = Bdg , d ≥ 2g − 1, follow from the relation Ag2g−1 = B2g−1 . In Section 4.3 we prove that the A- and the B-class behave in the same way upon the pullback along the forgetful map. We then use this result in Section 4.4 in order to show that Conjecture 2.5 is true if and only if it is true when all di ’s are positive. In Section 4.5 we prove that the classes Agd1 ,...,dn ,1 and Bdg1 ,...,dn ,1 behave in the same way upon the pushforward along the map forgetting the last marked point. Using this result, in Section 4.6 we show that Conjecture 2.5 is valid on Mg,n . In Section 4.7 we show that the conjectural relations (2.10) give a new formula for the class λg ∈ Rg (Mg ) and check the resulting formula for g ≤ 3. 4.1. Formulas with the double ramification cycles. First of all, let us recall the formula from [BSSZ15] for the product of the double ramification cycle with a ψ-class. Denote by glk : Mg1 ,n1 +k × Mg2 ,n2 +k → Mg1 +g2 +k−1,n1 +n2 the gluing map that corresponds to gluing a curve from Mg1 ,n1 +k to a curve from Mg2 ,n2 +k along the last k marked points on the first curve and the last k marked points on the second curve. Introduce the notation DRg1 (a1 , . . . , an ) ⊠k DRg2 (b1 , . . . , bm ) := =glk∗ (DRg1 (a1 , . . . , an ) × DRg2 (b1 , . . . , bm )) ∈ Rg1 +g2 +k (Mg1 +g2 +k−1,n+m−2k ). Let a1 , . . . , an be a list of integers with vanishing sum. For a subset I = {i1 , . . . , ik } ⊂ {1, . . . , n}, i1 < i2 < . . . < ik , let us denote by AI the list ai1 , . . . , aik and by aI the sum

DR/DZ equivalence conjecture and tautological relations

P

i∈I

17

ai . Assume that as 6= 0 for some 1 ≤ s ≤ n. Then we have [BSSZ15, Theorem 4]

(4.1)

=

as ψs DRg (a1 , . . . , an ) = X X X

X

ρ r ≥1

g1 ,g2 ≥0 kP I⊔J={1,...,n} p≥1 1 ,...,kp g1 +g2 +p−1=g aI >0 kj =aI

where r := 2g − 2 + n and ρ :=

(

Qp

i=1

ki

p!

DRg1 (AI , −k1 , . . . , −kp ) ⊠p DRg2 (AJ , k1 , . . . , kp ),

2g2 − 2 + |J| + p, if s ∈ I; −(2g1 − 2 + |I| + p), if s ∈ J.

Let us also recall the formula for the intersection of the double ramification cycle with a boundary divisor on Mg,n . For 0 ≤ h ≤ g and a subset I ⊂ {1, . . . , n} we have [BSSZ15] δhI · DRg (a1 , . . . , an ) = DRh (AI , −aI ) ⊠1 DRg−h (AI c , aI ) , where I c := {1, 2, . . . , n}\I. 4.2. One-point case. Lemma 4.1. Let g ≥ 1. Then for any k ≥ 0 we have Ag2g−1+k = ψ1k Ag2g−1 . Proof. Let π : Mg,2 → Mg,1 be the forgetful map that forgets the second marked point. We compute f = π∗ (π ∗ ψ1 · λg DRg (a, −a)) = aψ1 Ag,1 (a) =ψ1 λg DRg (a, −a)   {1,2} =π∗ (ψ1 · λg DRg (a, −a)) − π∗ δ0 · λg DRg (a, −a) = X g2 = π∗ (λg DRg1 (a, −a) ⊠1 DRg2 (−a, a)) − π∗ (λg DRg (0) ⊠1 DR0 (a, −a, 0)) = g g ,g ≥1 1

2

g1 +g2 =g

=Ag,2 (a), where we used that λg DRg (0) = (−1)g λ2g = 0. If k = 1, then the lemma is proved. If k ≥ 2, then we write the equation (a1 ψ1 )k Ag,1 (a) = (a1 ψ1 )k−1 Ag,2 (a) and apply formula (4.1) to the right-hand side of it k − 1 times. The lemma is proved.  On the other hand, it is not hard to get an explicit expression for the class Bdg . Let P g1 ,...,gk g1 , g2, . . . , gk ≥ 1 and d1 , . . . , dk ≥ 0. Introduce a class Cd1 ,...,dk ∈ R di +k−1(MP gi ,1 ) by ,...,gk Cdg11,...,d := k

g1 ψ

d1

g2 ψ

d2

...

gk ψ

dk

.

Then it is easy to see that for g ≥ 1 and d ≥ 2g − 1 we have Bdg

=

g X

X

X

,...,gk (−1)k−1 Cdg11,...,d , k

,...,gk ≥1 d1 ,...,dk k=1 g1P gi =g

where the last sum is taken over all non-negative integers d1 , . . . , dk satisfying l X i=1

k X i=1

di + l − 1 ≤ 2

l X i=1

di + k − 1 = d.

gi − 2,

if 1 ≤ l ≤ g − 1,

18

A. Buryak, J. Gu´er´e, P. Rossi

g We see that Bdg = ψ1d−2g+1 B2g−1 . Thus, for n = 1 Conjecture 2.5 is equivalent to the sequence of relations g Ag2g−1 = B2g−1 , g ≥ 1.

4.3. String equation. In this section we prove that the A- and the B-class behave in the same way upon the pullback along the forgetful map π : Mg,n+1 → Mg,n . Proposition 4.2. Denote by π : Mg,n+1 → Mg,n the forgetful map that forgets the last marked point. Then we have ( ∗ g P π A , if di = 2g − 1, d ,...,d n 1 P P Agd1 ,...,dn ,0 = (4.2) {i,n+1} ∗ g ∗ g π Ad1 ,...,dn + 1≤i≤n δ0 π Ad1 ,...,di −1,...,dn , if di ≥ 2g. Proof. Let m :=

(4.3)

P

di ≥1

di − 2g + 2. The proposition is equivalent to the equation (

eg,m (a1 , . . . , an ), π∗A if m = 1, Pn {i,n+1} ∗ eg,m−1 ∗ eg,m π A (a1 , . . . , an ) + i=1 ai δ0 π A (a1 , . . . , an ), if m ≥ 2, P bg,m (a0 , a1 , . . . , an ) where a1 , . . . , an are arbitrary integers. Let a0 := − ni=1 ai . Introduce a class A by X bg,m (a0 , a1 , . . . , an ) := a(Γ)λg DRΓ (a0 , a1 , . . . , an ) . A eg,m (a1 , . . . , an , 0) = A

Γ∈STm g,n+1

Formula (4.3) follows from the equation

(4.4) bg,m (a0 , . . . , an , 0) = A

(

bg,m (a0 , . . . , an ), π∗A if m = 1, Pn {i,n+1} ∗ bg,m−1 ∗ bg,m π A (a0 , . . . , an ), if m ≥ 2, π A (a0 , . . . , an ) + i=1 ai δ0

where the map π : Mg,n+2 → Mg,n+1 forgets the last marked point. For m = 1 equation (4.4) is clear. Suppose that m ≥ 2. Consider a stable tree Γ ∈ STm g,n+2 . Recall that we denote by li (Γ) the leg of Γ marked by 0 ≤ i ≤ n + 1. We will call a vertex v ∈ V (Γ) exceptional, if g(v) = 0, n(v) = 3 and the leg ln+1 (Γ) is incident to v. An exceptional vertex v ∈ V (Γ) will be called bad, if it is not incident to any leg li (Γ), where 1 ≤ i ≤ n. We will call the tree Γ bad, if it has a bad vertex. Otherwise, it will be called good. For a vertex v ∈ V (Γ) let ( 2g(v) + n(v) − 2, if ln+1 (Γ) is not incident to v, 2g(v) + n(v) − 3, if ln+1 (Γ) is incident to v.

r ′ (v) :=

′ For a good stable tree Γ ∈ STm g,n+2 introduce a constant a (Γ) by   Y Y r ′(v) P a′ (Γ) :=  a(h) e (Γ) h∈H+

v∈V (Γ) v is not exceptional

v∈Desc[v] e

r ′ (e v)

.

Using these notations, we can rewrite the right-hand side of (4.4) as follows: bg,m (a0 , . . . , an )+ π A ∗

n X i=1

{i,n+1} ∗

ai δ0

bg,m−1 (a0 , . . . , an ) = π A

On the other hand, by definition, bg,m (a0 , . . . , an , 0) = A

X

Γ∈STm g,n+2

X

a′ (Γ)λg DRΓ (a0 , a1 , . . . , an , 0) .

Γ∈STm g,n+2 Γ is good

a(Γ)λg DRΓ (a0 , a1 , . . . , an , 0) .

DR/DZ equivalence conjecture and tautological relations L′

r1

...

A1

rk−1

z .}| .. { rk

.. .

Ak−1

ln+1

l rk+1

1 rk+1

19

A1k+1

Alk+1

Figure 1. Stable tree Γ We see that we have to prove the equation X a(Γ)λg DRΓ (a0 , a1 , . . . , an , 0) = (4.5) Γ∈STm g,n+2

X

a′ (Γ)λg DRΓ (a0 , a1 , . . . , an , 0) .

Γ∈STm g,n+2 Γ is good

Let us prove equation (4.5). Suppose Γ is a bad stable tree. Let us show how to express the class a(Γ)λg DRΓ (a0 , a1 , . . . , an , 0) as a linear combination of the classes λg DRΓe (a0 , a1 , . . . , an , 0), e are good. Suppose that s ≥ 2 and b1 , . . . , bs are integers with vanishing where the stable trees Γ sum. We have the following relation in the cohomology of Mg,s+2 (see e.g. [Bur15, eq. (5.2)]): X X λg (4.6) bI DRg1 (0, BI , −bI ) ⊠1 DRg2 (0, BJ , −bJ ) = 0. I⊔J={1,...,s} g1 +g2 =g I,J6=∅

Suppose that the point with the zero multiplicity in the second double ramification cycle is marked by s + 2. Let us multiply relation (4.6) by ψs+2 and push it forward to Mg,s+1 by forgetting the last marked point: X X (4.7) bI (2g2 + |J| − 1)DRg1 (0, BI , −bI ) ⊠1 DRg2 (BJ , −bJ ) = 0. λg I⊔J={1,...,s} g1 +g2 =g 2g2 +|J|−1>0 I,J6=∅

Suppose that the level of the bad vertex in our bad stable tree Γ is equal to k. Then relation (4.7) allows to express the class a(Γ)λg DRΓ (a0 , . . . , an , 0) in terms of the classes λg DRΓe (a0 , . . . , an , 0), e is good or bad with the bad vertex of level k + 1. Therefore, applying where the tree Γ relation (4.7) sufficiently many times, we come to a decomposition X e g DRe (a0 , . . . , an , 0), a(Γ, Γ)λ a(Γ)λg DRΓ (a0 , . . . , an , 0) = Γ m e Γ∈ST g,n+2 e is good Γ

e are certain coefficients. We see that for any good graph Γ we have to prove the where a(Γ, Γ) identity X e Γ) = a′ (Γ). (4.8) a(Γ, a(Γ) + m e Γ∈ST g,n+2 e is bad Γ

Let us prove (4.8). Suppose that the leg ln+1 = ln+1 (Γ) is incident to a vertex of level k. Denote it by vk . Denote by v1 the root of Γ. Let v1 , v2 , . . . , vk be the unique path connecting v1 1 l and vk . Denote by vk+1 , . . . , vk+1 , l ≥ 0, the direct descendants of vk . Let L′ := L[vk ]\{ln+1 }. In Fig. 1 we draw our tree Γ. Note that each vertex v in the picture is decorated by the number r(v), instead of the genus. This is more convenient for the computations. We use the j j notations ri := r(vi ), 1 ≤ i ≤ k, and rk+1 := r(vk+1 ), 1 ≤ j ≤ l. The symbols Ai and Ajk+1 j indicate the pieces of the tree Γ that don’t contain the vertices vi and vk+1 . Let us also introduce

20

A. Buryak, J. Gu´er´e, P. Rossi L′

r1

...

A1

ri−1

1

ri

Ai−1

ln+1

Ai

rk−2

z .}| .. {

Ak−2

Ak−1

...

re

1 rk+1

A1k+1

.. . l rk+1

Alk+1

1 rk+1

A1k+1

Figure 2. Bad stable tree of the first type

L′

r1 A1

...

ri−1 Ai−1

...

ri

1 ln+1

rk−1

z .}| .. {

Ak−1

Ai

re

.. . Ajk+1 .. . l rk+1

Alk+1

Figure 3. Bad stable tree of the second type the following notations: X

Ri :=

r(v),

1 ≤ i ≤ k,

v∈Desc[vi ] j Rk+1

:=

X

r(v),

1 ≤ j ≤ l,

j ] v∈Desc[vk+1

e a := a(Γ)

,

k l j Y ri Y rk+1 j Ri j=1 Rk+1 i=1

!

There are two cases: the vertex vk can be exceptional or not. Suppose that vk is not exceptional. Then a′ (Γ) = e a

l j r1 · · · rk−1 (rk − 1) Y rk+1 . j (R1 − 1) · · · (Rk − 1) j=1 Rk+1

e such that a(Γ, e Γ) 6= 0. These It is not hard to understand the structure of bad stable trees Γ trees are of two types. A bad tree of of the first type is shown in Fig. 2, where 1 ≤ i ≤ k − 1 and re = rk−1 + rk − 1. A bad tree of the second type is shown in Fig. 3, where 1 ≤ i ≤ k, j 1 ≤ j ≤ l and re = rk + rk+1 − 1. It is not hard to see that  j Ql rk+1 ···rk−1 e a R1 ···Ri (Rri1−1)···(R , j=1 −1) Rjk+1 k−1 e Γ) = a(Γ, Ql r1 ···rk−1 j −e a R1 ···Ri (R Rk+1 m=1 i −1)···(Rk −1)

m rk+1 Rm k+1

e is of the first type, if Γ

e is of the second type. , if Γ

DR/DZ equivalence conjecture and tautological relations

21

Therefore, equation (4.8) follows from the identity k k−1 k X l Y X ri X r1 · · · rk−1 r1 · · · rk−1 j + − Rk+1 = R R · · · R (R − 1) · · · (R − 1) R · · · R (R − 1) · · · (R − 1) i 1 i i k−1 1 i i k i=1 i=1 i=1 j=1

=

r1 · · · rk−1 (rk − 1) , (R1 − 1) · · · (Rk − 1)

or, equivalently, k−1

k

X X 1 Rk − rk rk + − = (4.9) R1 · · · Rk R1 · · · Ri (Ri − 1) · · · (Rk−1 − 1) i=1 R1 · · · Ri (Ri − 1) · · · (Rk − 1) i=1 =

rk − 1 . (R1 − 1) · · · (Rk − 1)

Note that rk Rk − rk rk − 1 − = , R1 · · · Rk R1 · · · Rk (Rk − 1) R1 · · · Rk−1 (Rk − 1) 1 Rk − rk rk − 1 − = , R1 · · · Ri (Ri − 1) · · · (Rk−1 − 1) R1 · · · Ri (Ri − 1) · · · (Rk − 1) R1 · · · Ri (Ri − 1) · · · (Rk − 1) where 1 ≤ i ≤ k − 1. Therefore, equation (4.9) is equivalent to the equation k−1

(4.10)

X 1 1 1 + = , R1 · · · Rk−1 i=1 R1 · · · Ri (Ri − 1) · · · (Rk−1 − 1) (R1 − 1) · · · (Rk−1 − 1)

which can be easily proved by induction on k. Suppose that vk is exceptional. Then l = 0, the set L′ consists of only one leg and rk = Rk = 1. We have r1 · · · rk−1 . a′ (Γ) = e a (R1 − 1) · · · (Rk−1 − 1) e with a(Γ, e Γ) 6= 0 should necessarily be of the first type (see Fig. 2) and A bad stable tree Γ then we have r1 · · · rk−1 e Γ) = e a(Γ, a . R1 · · · Ri (Ri − 1) · · · (Rk−1 − 1)

We immediately see that again equation (4.8) follows from the elementary identity (4.10). The proposition is proved. 

Proposition 4.3. Denote by π : Mg,n+1 → Mg,n the forgetful map that forgets the last marked point. Then we have P ( ∗ g π Bd1 ,...,dn , if di = 2g − 1, P P {i,n+1} ∗ g (4.11) Bdg1 ,...,dn ,0 = ∗ g π Bd1 ,...,dn + 1≤i≤n δ0 π Bd1 ,...,di −1,...,dn , if di ≥ 2g. di ≥1

Proof. Let (T, q) ∈ ΩB,g d1 ,...,dn be an admissible and stable complete tree with a power function q : H+em (T ) → N,

as in Definition 2.4. We denote by deg(T ) its number of levels. In particular, there are extra legs at every vertex (except the root) that we will eventually forget when computing the B-class. C C C Choose a vertex v ∈ V (T ). Let C = (eC 1 , v1 , . . . , edeg(T )−l(v) , vdeg(T )−l(v) , σn+1 ) be a chain of weakly stable vertices with a new marking σn+1 . Precisely, the edge eC 1 is attached to C C the vertex v1C , the edge eC links the vertex v to v , and the leg σ is attached to the n+1 k k−1 k C vertex vdeg(T )−l(v) . Moreover, every vertex is of genus 0 and contains an extra leg. We construct a tree Tv , obtained from T by gluing the edge eC 1 (and thus the chain C) to the vertex v. We

22

A. Buryak, J. Gu´er´e, P. Rossi

have H+em (T ) ⊂ H+em (Tv ) and we extend the power function q into a function qv : H+em (Tv ) → N by taking qv (hC k ) := 0 and qv (σn+1 ) := 0, em C where hC k is the half-edge in H+ (Tv ) contained in the edge ek . It is easy to see that we get (Tv , qv ) ∈ ΩB,g d1 ,...,dn ,0 . Choose a half-edge h ∈ H+em (T ) attached to the vertex v and such that q(h) > 0. We construct a tree T(v,h) , obtained from T by adding an extra level between the levels l(v) and l(v) + 1 of T as follows: - denote by h0 , . . . , hm ∈ H+em (T ) the half-edges of level l(v), with h0 := h, - insert a pair (ek , vk ) between the half-edge hk and the vertex it is attached to, where ek = (h′k , h′′k ) is an edge and vk is a vertex of genus 0, - glue the half-edge hC 1 from the chain C to the vertex v0 , - add q(h) extra legs to the vertex v0 and q(hk )+1 extra leg to the vertex vk , for 1 ≤ k ≤ m. Therefore, the number of levels of the tree T(v,h) is deg(T ) + 1, the vertex v0 ∈ V (T(v,h) ) is the only strongly stable vertex at its level, and we have a natural inclusion H+em(Tv ) ⊂ H+em (T(v,h) ). Then, we extend the power function qv into a function q(v,h) : H+em (T(v,h) ) → N by taking  q(hk ) − 1, if k = 0, ′ q(v,h) (hk ) := q(hk ), if k 6= 0. The complete tree T(v,h) is obviously stable, but not necessarily admissible. We get (T(v,h) , q(v,h) ) ∈

ΩB,g d1 ,...,dn ,0

⇐⇒ l(v) 6= deg(T ) or

n X

di = 2g − 1.

i=1

Furthermore, observe that when l(v) = deg(T ), then the half-edge h corresponds to a marking σi and we get {i,n+1}

e∗ [T(v,h) , q(v,h) ] = σi∗ e∗ [T, qi ] = δ0

· π ∗ e∗ [T, qi ] ∈ R∗ (Mg,n+1 ),

where the morphism σi denotes here the section of the i-th marking in the universal curve Cg,n ≃ Mg,n+1 , and where qi : H+em(T ) → N is defined by  di − 1, if h = σi , qi (h) := q(h), otherwise.

Conversely, let (T ′ , q ′ ) ∈ ΩB,g d1 ,...,dn ,0 and denote by v the first strongly stable ancestor of the marking σn+1 . In particular, the marking σn+1 is attached to the vertex v via a chain C of weakly stable vertices and we denote by h(n+1) ∈ H+em(T ′ ) the half-edge from C attached to v. We have two possibilities: (1) v is a vertex of genus 0 with exactly two half-edges h, h(n+1) ∈ H+em (T ′ ) attached to it and v is the only strongly stable vertex of level l(v), (2) v is another type of vertex. Denote by T the tree obtained from T ′ by forgetting the chain C containing the marking σn+1 , and contracting the level l(v) in case (1). In particular, the power function q ′ restricts to a function q and we get  (T(v,h) , q(v,h) ) in case (1), B,g ′ ′ (T, q) ∈ Ωd1 ,...,dn and (T , q ) = (Tv , qv ) in case (2). Furthermore, from the formula

(4.12)

π∗

v q1 ···qr

=

−

v q1 ···qr

n+1

X

v qi −1

1≤i≤r qi >0 q1 ···qˆi ···qr

0 n+1 i

DR/DZ equivalence conjecture and tautological relations

expressing the pull-back of ψ-classes via the map π, we obtain  e∗ π ∗ ([T, q]) =

X v∈T

 e∗  [Tv , qv ] −

X

23



 [T(v,h) , q(v,h) ] ,

em (T ) h∈H+ h→v,q(h)>0

for every (T, q) ∈ ΩB,g d1 ,...,dn and where h → v means that the half-edge h is incident to the vertex v. As a consequence, when d1 + · · · + dn ≥ 2g, we obtain X π ∗ (Bdg1 ,...,dn ) = (−1)deg(T )−1 π ∗ e∗ [T, q] (T,q)∈ΩB,g d ,...,dn 1

X

=

(−1)deg(T )−1 e∗ π ∗ [T, q],

(T,q)∈ΩB,g d ,...,dn 1

where the second equality comes from the general fact that Mg,n+2 is birational to the fiber product Mg,n+1 ×Mg,n Mg,n+1, and then   X

π ∗ (Bdg1 ,...,dn ) =

(−1)deg(T )−1

v∈T

(T,q)∈ΩB,g d ,...,dn 1

X

=

X e∗ [Tv , qv ] − 

(−1)deg(T

′ )−1

(T ′ ,q ′ )∈ΩB,g d ,...,dn ,0 1

−

X

(−1)deg(T )−1 X

X

1

{i,n+1}

δ0

e∗ [T ′ , q ′ ] X

(−1)deg(T )−1

(T,q)∈ΩB,g d ,...,dn

= Bdg1 ,...,dn ,0 −

em (T ) h∈H+ h→v,q(h)>0

X

 e∗ [T(v,h) , q(v,h) ] 

e∗ [T(v,h) , q(v,h) ]

em (T ) h∈H+

v∈T l(v)=deg(T ) h→v,q(h)>0

(T,q)∈ΩB,g d1 ,...,dn

= Bdg1 ,...,dn ,0 −

X

X

{i,n+1}

δ0

· π ∗ e∗ [T, qi ]

1≤i≤n di >0

· π ∗ Bdg1 ,...,di −1,...,dn .

1≤i≤n di >0

When d1 +· · ·+dn = 2g −1, then we have seen that (T(v,h) , q(v,h) ) is always admissible, so that the first three equalities are the same, but there is no second term in the last three equalities. Hence we get π ∗ Bdg1 ,...,dn = Bdg1 ,...,dn ,0 .  4.4. Reduction of the conjecture. Proposition 4.4. Conjecture 2.5 is true if and only if it is true when all di ’s are positive. Furthermore, Conjecture 2.5 is true in genus 0 and in genus 1. Proof. The first statement follows immediately from Propositions 4.2 and 4.3. Assume g = 0. Since dim M0,n = n − 3, the classes A0d1 ,...,dn and Bd01 ,...,dn are non-trivial only P if di ≤ n − 3. Therefore, we can always apply formulas (4.2) and (4.11) to them, unless n = 3 and d1 = d2 = d3 = 0, where we get 0 A00,0,0 = B0,0,0 = 1 ∈ H 0 (M0,3 , Q).

24

A. Buryak, J. Gu´er´e, P. Rossi L′

r1′′

r′

z .}| .. { r

.. .

A′

l1

rl′′

A′′1

A′′l

Figure 4. Stable tree Γ 1 1 Assume P g = 1. Since dim M1,n = n, the classes Ad1 ,...,dn and Bd1 ,...,dn are non-trivial only if di ≤ n. Therefore, we can always apply formulas (4.2) and (4.11) to them, un1 less d1 = d2 = . . . = dn = 1. In order to prove that A11,1,...,1 = B1,1,...,1 , it is sufficient to R R 1 1 check that M1,n A1,1,...,1 = M1,n B1,1,...,1 . Note that these two integrals are equal to hτ1 (e1 )n iDR 1

DR = F red and hτ1 (e1 )n ired 1 , respectively, for the trivial cohomological field theory. The equality F for the trivial cohomological field theory was checked in [BDGR16a]. Therefore, Conjecture 2.5 is true in genus 1. 

4.5. Dilaton equation. Here we prove that the classes Agd1 ,...,dn ,1 and Bdg1 ,...,dn ,1 behave in the same way upon the pushforward along the map forgetting the last marked point. Proposition 4.5. Denote by π : Mg,n+1 → Mg,n the forgetful map that forgets the last marked point. Then we have ( P (2g − 2 + n)Agd1 ,...,dn , if di > 2g − 2, g P π∗ (Ad1 ,...,dn ,1 ) = (4.13) 0, if di = 2g − 2. Before proving the proposition let us formulate three auxiliary statements. Recall that for a stable tree Γ ∈ STm g,n+1 we denote by v1 (Γ) the root of Γ and by li (Γ), 0 ≤ i ≤ n, the leg of Γ marked by i. Lemma 4.6. Let a0 , . . . , an , n ≥ 1, be integers with vanishing sum and m ≥ 2. Then we have X 2g − 1 + n bg,m (a0 , . . . , an ) − a1 ψ1 A bg,m−1 (a0 , . . . , an ) = a(Γ)λg DRΓ (a0 , . . . , an ). A r(v1 (Γ)) Γ∈STm g,n+1

v(l1 (Γ))=v1 (Γ)

Proof. Using formula (4.1), for an arbitrary stable tree Γ ∈ STm−1 g,n+1 we can write a decomposition X e g DRe (a0 , . . . , an ), a1 ψ1 · a(Γ)λg DRΓ (a0 , . . . , an ) = a(Γ, Γ)λ Γ m e Γ∈ST g,n+1

e are certain coefficients. Let Γ ∈ STm . The statement of the lemma is equivawhere a(Γ, Γ) g,n+1 lent to the following equation: ( 2g−1+n X , if l1 (Γ) is incident to v1 (Γ), e (4.14) a(Γ) − a(Γ, Γ) = r(v1 (Γ)) 0, otherwise. m−1 e Γ∈STg,n+1

Let v ∈ V (Γ) be the vertex incident to l1 = l1 (Γ). Denote by v1′′ , . . . , vl′′P , l ≥ 0, the direct ′ ′′ ′′ descendants of v. Let L := L[v]\{l1 }, r := r(v), ri := r(vi ), R := v ) and v∈Desc[v] r(e e P Ri′′ := ve∈Desc[v′′ ] r(e v). i Suppose that v 6= v1 (Γ). Denote by v ′ ∈ V (Γ) the mother of v and let r ′ := r(v ′ ) and P R′ := ve∈Desc[v′ ] r(e v). We draw the stable tree Γ in Fig. 4. Similarly to the figures in the proof of Proposition 4.2, we decorate a vertex w of Γ by number r(w). It is not hard to see that there

DR/DZ equivalence conjecture and tautological relations

L′

l1

z .}| .. {

.. .

r ′ +r

A′

z .}| .. {

r′

rl′′

r1′′

L′

A′′1

r1′′

A′′l

25

r+rj′′

A′′1 .. .

A′′j .. .

l1

A′

rl′′

A′′l

e such that a(Γ, e Γ) 6= 0 Figure 5. Stable trees Γ L′

z .}| .. {

r1′′

r

.. .

l1

rl′′

r1′′

L′

z .}| .. {

A′′1

.. . A′′j

r+rj′′

A′′l

A′′1

.. .

l1 rl′′

A′′l

Γ e Γ

e such that a(Γ, e Γ) 6= 0 Figure 6. Stable tree Γ and stable trees Γ

e ∈ STm−1 such that a(Γ, e Γ) 6= 0. The first one is shown on the are exactly l + 1 stable trees Γ g,n+1 left-hand side of Fig. 5, and the other l trees are on the right-hand side, where 1 ≤ j ≤ l. Let e a := a(Γ)

,

l r ′ r Y rj′′ R′ R j=1 Rj′′

!

.

e Γ) for the left tree in Fig. 5 is equal to e The coefficient a(Γ, a Rr ′ r ′ R′′ Q r ′′ in Fig. 5 it is equal to −e a R′ Rj lk=1 Rk′′ . We compute ′

k

X

m−1 e Γ∈ST g,n+1

l

e Γ) = e a(Γ, a

X r ′Rj′′ r′ − R′ j=1 R′ R

!

Ql

rk′′ k=1 R′′ k

and for the right tree

l l Y rk′′ r ′ r Y rk′′ =e a ′ = a(Γ). Rk′′ RR Rk′′

k=1

k=1

Therefore, formula (4.14) is proved in the case when l1 is not incident to v1 (Γ). e such that a(Γ, e Γ) 6= 0 are shown in Suppose that v = v1 (Γ). The tree Γ and stable trees Γ Fig. 6. Let ! , l r Y rj′′ . e a := a(Γ) R j=1 Rj′′

26

A. Buryak, J. Gu´er´e, P. Rossi R′′

e Γ) for the right tree in Fig. 6 is equal to −e The coefficient a(Γ, a Rj a(Γ) −

X

m−1 e Γ∈ST g,n+1

The lemma is proved.

l

e Γ) = e a(Γ, a

r X Rj′′ + R j=1 R

!

Ql

rk′′ . k=1 R′′ k

So we compute

l l Y Y rk′′ rk′′ R = e a = a(Γ). ′′ ′′ Rk Rk r k=1 k=1



Lemma 4.7. Let a0 , . . . , an , n ≥ 1, be integers with vanishing sum and m ≥ 2. Then we have X 2g − 1 + n bg,m (a0 , . . . , an ) − a0 ψ0 A bg,m−1 (a0 , . . . , an ) = a(Γ)λg DRΓ (a0 , . . . , an ). A r(v 1 (Γ)) m Γ∈ST g,n+1

Proof. The proof is analogous to the proof of the previous lemma.



Corollary 4.8. Let a1 , . . . , an , n ≥ 1, be arbitrary integers and m ≥ 2. Denote by π : Mg,n+1 → Mg,n the forgetful map that forgets the first marked point. Then we have (4.15) Ag,m (a1 , . . . , an ) − a1 ψ1 Ag,m−1 (a1 , . . . , an ) =  X  X 2g − 1 + n a(Γ) bg,m−1 P λg π∗ DRΓ − ai , a1 , . . . , an +A = ai r(v1 (Γ)) m Γ∈STg,n+1 v(l1 (Γ))=v1 (Γ) g(v1 (Γ))≥1

−

n X

!

ai , a2 , . . . , an .

i=2

Proof. The corollary is an elementary exercise that uses two previous lemmas and the fact that  X  1 g,m g,m b P A (a1 , . . . , an ) = − ai , a1 , . . . , an . π∗ A ai



Proof of Proposition 4.5 Let m := (4.16)

P

di − 2g + 3. Let us prove that ( 0, if m = 1, ∂ = π∗ Ag,m (a1 , . . . , an+1 ) g,m−1 ∂an+1 π∗ (ψn+1 A (a1 , . . . , an , 0)) , if m ≥ 2. an+1 =0

For m = 1 this equation immediately follows from Lemma 2.1. Suppose m ≥ 2. Let us rewrite equation (4.15) in the way that is more suitable for us: (4.17) Ag,m (a1 , . . . , an+1 ) − an+1 ψn+1 Ag,m−1 (a1 , . . . , an+1 ) = ! n  X  X X 2g + n a(Γ) bg,m−1 − P λg π0∗ DRΓ − = ai , a1 , . . . , an , ai , a1 , . . . , an+1 +A r(v1 (Γ)) ai i=1 Γ∈STm g,n+2

v(ln+1 (Γ))=v1 (Γ) g(v1 (Γ))≥1

where the map π0 : Mg,n+2 → Mg,n+1 forgets the first marked point. The last term on the right-hand side of this equation doesn’t depend on an+1 . Note also that, by Lemma 2.1, after applying the push-forward π∗ each term in the sum on the right-hand side of (4.17) becomes divisible by a2n+1 . This proves equation (4.16). Equation (4.16) immediately implies the statement of the proposition for m = 1. In the case m ≥ 2 equation (4.16) yields
by Prop. 4.2
π∗ Agd1 ,...,dn ,1 = π∗ ψn+1 Agd1 ,...,dn ,0 = π∗ ψn+1 π ∗ Agd1 ,...,dn = (2g − 2 + n)Agd1 ,...,dn . The proposition is proved.



DR/DZ equivalence conjecture and tautological relations

27

Proposition 4.9. Denote by π : Mg,n+1 → Mg,n the forgetful map that forgets the last marked point. Then we have ( P (2g − 2 + n) Bdg1 ,...,dn , if di > 2g − 2, g P π∗ (Bd1 ,...,dn ,1 ) = (4.18) 0, if di = 2g − 2. Proof. Let (T, q) ∈ ΩB,g d1 ,...,dn be an admissible and stable complete tree with a power function q : H+em (T ) → N, as in Definition 2.4. We denote by ǫ : {1, . . . , deg(T ) − 1} → N the function        X −2− g(v) ǫ(k) := 2    

X

em (T ) h∈H+ l(h)=k

v∈V (T ) l(v)≤k

 q(h) 

measuring the distance to non-admissibility at the level k. As in the proof of Proposition 4.3, we have two possible ways to add a new marking labelled by n + 1. C C C First, choose a vertex v ∈ V (T ). Let C = (eC 1 , v1 , . . . , edeg(T )−l(v) , vdeg(T )−l(v) , σn+1 ) be a chain of weakly stable vertices with a new marking σn+1 . Precisely, the edge eC 1 is attached C C to the vertex v1C , the edge eC links the vertex v to v , and the leg σ is attached to n+1 k k−1 k C the vertex vdeg(T )−l(v) . Moreover, every vertex is of genus 0 and contains two extra legs. We construct a tree Tv , obtained from T by gluing the edge eC 1 (and thus the chain C) to the vertex v. We have H+em (T ) ⊂ H+em (Tv ) and we extend the power function q into a function qv : H+em (Tv ) → N by taking qv (hC k ) := 1 and qv (σn+1 ) := 1, em C where hC k is the half-edge in H+ (Tv ) contained in the edge ek . It is easy to see that we get

(Tv , qv ) ∈ ΩB,g d1 ,...,dn ,1 ⇐⇒ ∀k ∈ [l(v), deg(T ) − 1], ǫ(k) ≥ 1. In particular, when the vertex v is at the maximal level deg(T ), then the tree Tv is always admissible. Second, choose a half-edge h ∈ H+em (T ) attached to the vertex v. We construct a tree T(v,h) , obtained from T by adding an extra level between the levels l(v) and l(v) + 1 of T as follows: - denote by h0 , . . . , hm ∈ H+em (T ) the half-edges of level l(v), with h0 := h, - insert a pair (ek , vk ) between the half-edge hk and the vertex it is attached to, where ek = (h′k , h′′k ) is an edge and vk is a vertex of genus 0, - glue the half-edge hC 1 from the chain C to the vertex v0 , - add q(hk ) + 1 extra legs to the vertex vk , for 0 ≤ k ≤ m. Therefore, the number of levels of the tree T(v,h) is deg(T ) + 1, the vertex v0 ∈ V (T(v,h) ) is the only strongly stable vertex at its level, and we have a natural inclusion H+em(Tv ) ⊂ H+em (T(v,h) ). Then, we extend the power function qv into a function q(v,h) : H+em (T(v,h) ) → N by taking q(v,h) (h′k ) := q(hk ). We obtain (T(v,h) , q(v,h) ) ∈

ΩB,g d1 ,...,dn ,1

⇐⇒



∀k ∈ [l(v), deg(T ) − ǫ(k) ≥ 1, and P1], n (l(v) 6= deg(T ) or i=1 di = 2g − 2) .

In particular, when the vertex v is at the maximal level deg(T ), the tree T(v,h) is admissible if and only if d1 + · · · + dn = 2g − 2. Let lT ∈ [1, deg(T )] be the smallest integer such that ∀k ∈ [lT , deg(T ) − 1], ǫ(k) ≥ 1.

28

A. Buryak, J. Gu´er´e, P. Rossi

When d1 + · · · + dn > 2g − 2 (resp. when d1 + · · · + dn = 2g − 2), the two constructions (T, q, v) 7→ (Tv , qv ) and (T, q, v, h) 7→ (T(v,h) , q(v,h) ) give a bijection from the set G {v ∈ V (T )|l(v) ≥ lT } ⊔ {(v, h) ∈ V (T ) × H+em (T )|h → v and lT ≤ l(v) < deg(T )} (T,q)∈ΩB,g d ,...,dn 1

(resp. the same set with the inequality lT ≤ l(v) ≤ deg(T )) to the set ΩB,g d1 ,...,dn ,1 . Furthermore, we get the contributions (4.19) (4.20)

e∗ π∗ ([Tv , qv ]) = (2g(v) − 2 + n(v) + q(v) + 1)e∗ [T, q], e∗ π∗ ([T(v,h) , q(v,h) ]) = (q(h) + 1)e∗ [T, q],

where q(v) denotes the value of the power function q : H+em(T ) → N at the (half-)edge linking the mother of the vertex v to the vertex v, and n(v) denotes the number of half-edges attached to the vertex v, without counting the extra legs. Thus, the total number of half-edges attached to the vertex v is indeed n(v) + q(v) + 1. Finally, when d1 + · · · + dn > 2g − 2, we get X (−1)deg(T )−1 π∗ e∗ [T, q] π∗ (Bdg1 ,...,dn ,1 ) = (T,q)∈ΩB,g d ,...,dn ,1 1

=

X

(T,q)∈ΩB,g d ,...,dn 1



 (−1)deg(T )−1  

v∈V (T ) l(v)=deg(T )

X

+(−1)deg(T )−1

v∈V (T ) lT ≤l(v)0


aJ f AI , aJ ⊠1 DRg2 (AJ , −aJ ) . DRg1 −a, a

Let us prove now that A2d1 ,d2 = Bd21 ,d2 , where (d1 , d2 ) = (3, 1) or (d1 , d2) = (2, 2). By equation (A.8), we have   1 2 2 ^ Ad1 ,d2 − ψ1 Ad1 −1,d2 =Coef ad1 ad2 λ2 DR1 (−a = 1 − a2 , a1 , a2 ) ⊠1 DR1 (a2 , −a2 ) 1 2 a1 + a2 ! (a1 + a2 )a32 0 0 0 =Coef ad1 ad2 = 0. 1 2 576 1 2 On the other hand, it is easy to compute that 2 B3,1 =ψ13 ψ2 −

1

2 B2,2 =ψ12 ψ22 −

1

ψ2 1 1 ψ 2

−

ψ2 1 1 ψ 2

−

ψ3 1 1

1

,

2

1

ψ 1 ψ2

1

. 2

2 2 Comparing these expressions with formula (A.1), we can easily see that B3,1 = ψ1 B2,1 and 2 2 2 2 2 2 B2,2 = ψ1 B1,2 . Since the relation A2,1 = B2,1 is already checked, the relations A3,1 = B3,1 and 2 A22,2 = B2,2 are now also proved.

DR/DZ equivalence conjecture and tautological relations

39

2 A.5. Relation A22,1,1 = B2,1,1 . Using equation (A.8), we compute

A22,1,1 − ψ1 A21,1,1 = (A.9)

X

=

Coef a21 a2 a3

I⊔J={1,2,3} 1∈I, |J|≥1

(A.10) + Coef a21 a2 a3





 aJ λ2 DR1 (−a1 ^ − a2 − a3 , AI , aJ ) ⊠1 DR1 (AJ , −aJ ) + a1 + a2 + a3

 a2 + a3 λ2 DR2 (−a1 ^ − a2 − a3 , a1 , a2 + a3 ) ⊠1 DR0 (a2 , a3 , −a2 − a3 ) . a1 + a2 + a3

Let us look at a term in the sum in line (A.9). The class λ1 DR1 (AJ , −aJ ) is a polynomial in the variables aj , j ∈ J, and it doesn’t depend on a1 . We have 1 λ1 DR1 (−a1 ^ − a2 − a3 , AI , aJ ) = (a1 + a2 + a3 )λ1 ∈ R1 (M1,|I|+1). a1 + a2 + a3 So, the polynomial class in the brackets in line (A.9) depends on a1 at most linearly. Therefore, the expression in line (A.9) is equal to zero. Let us look at the expression in line (A.10). We can easily see that it is equal to  
b 2|0 2,0 ^ 2 · Coef a2 b2 (gl1|2,3 )∗ λ2 DR2 (−a − b, a, b) × [M0,3 ] = 2 · (gl1|2,3 )∗ A22,1 × [M0,3 ] . a+b As a result, we obtain


2|0 A22,1,1 = ψ1 A21,1,1 + 2 · (gl1|2,3 )∗ A22,1 × [M0,3 ] .

On the other hand, we have

2 B2,1,1 =ψ12 ψ2 ψ3 − 6

+6

1

1

2

ψ

ψ2

ψ

1

ψ

−3

0

2

ψ2

1

0

+3

1

1

+3

1

1

ψ

1 0 ψ

−

ψ ψ 1 ψ

1

−

ψ2 1 1

1 ψ

1

.

0 ψ

Using also formula (A.3), we compute 2 2 − ψ1 B1,1,1 =−3 B2,1,1

−2 =2

2

ψ2

1

1 0 ψ ψ 1 ψ

ψ 2 0 ψ2 1

+6

1

+2

ψ 2 0 ψ2 1

−2

1

1 ψ2

0 1

1 2

ψ2

0

0 1

ψ

1 0 ψ

=

0

−6 1

ψ

0

0

−2

1

ψ 1 ψ

0 1

−2

1

1 ψ2

0 1

.

1


2|0 2 Using (A.1) we see that the last expression is equal to 2 · (gl1|2,3 )∗ B2,1 × [M0,3 ] and we get
2|0 2 2 2 B2,1,1 = ψ1 B1,1,1 + 2 · (gl1|2,3 )∗ B2,1 × [M0,3 ] .

2 2 Since relations A22,1 = B2,1 and A21,1,1 = B1,1,1 are proved, we conclude that relation A22,1,1 = 2 B2,1,1 is true.

40

A. Buryak, J. Gu´er´e, P. Rossi

2 A.6. Relation A21,1,1,1 = B1,1,1,1 . We follow the same strategy, as in the previous section. Using equation (A.8), we compute

A21,1,1,1 − ψ1 A20,1,1,1 =  a X J f AI , aJ ) ⊠1 DR1 (AJ , −aJ ) + λ2 DR1 (−a, = Coef a1 a2 a3 a4 a

(A.11)

I⊔J={1,2,3,4} I∋1, |J|≥1

X

+

(A.12)

a

J

Coef a1 a2 a3 a4

a

I⊔J={1,2,3,4} I∋1, |J|≥2

 f λ2 DR2 (−a, AI , aJ ) ⊠1 DR0 (AJ , −aJ ) ,

P f AI , aJ ) = where a := 4i=1 ai . Let us look at a term in the sum in line (A.11). We have a1 λ1 DR1 (−a, 1 aλ1 ∈ R (M1,|I|+1) and the class λ1 DR1 (AJ , −aJ ) doesn’t depend on the variables ai , i ∈ I. Therefore, the coefficient of a1 a2 a3 a4 can be non-zero only if I = {1}. So the expression in line (A.11) is equal to 1|1

(gl1|2,3,4 )∗ (λ1 × Coef a2 a3 a4 ((a2 + a3 + a4 )λ1 DR1 (a2 , a3 , a4 , −a2 − a3 − a4 ))) = (A.13) = −

λ1 1

λ1 1

λ1 1

−3

0

1

λ1 1

λ1 1

+2

0

1

λ1 1

0

λ1 1

+3

1

The expression in line (A.12) is equal to
2|0 6(gl1|2,3,4 )∗ A21,2 × [M0,4 ] + 2

λ1 1

0

+3

λ1 1

1

X

λ1 1

0

.

1


2|0 (gl1,i|j,k )∗ A21,1,1 × [M0,3 ] .

{i,j,k}={2,3,4} j